Write the​ slope-intercept equation of the function f whose graph satisfies the given conditions. 
The graph of f passes through left parenthesis negative 6 comma 4 right parenthesis and is perpendicular to the line that has an​ x-intercept of 1 and a​ y-intercept of negative 3.

To find the slope-intercept equation of the function \( f \), we need to follow a few steps:

### Step 1: Determine the equation of the given line

We are told that the line has an \( x \)-intercept of 1 and a \( y \)-intercept of -3. These points are:

- \( (1, 0) \) for the \( x \)-intercept
- \( (0, -3) \) for the \( y \)-intercept

We can find the slope (\( m \)) of this line using the slope formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting the intercept points \( (0, -3) \) and \( (1, 0) \):
\[
m = \frac{0 - (-3)}{1 - 0} = \frac{3}{1} = 3
\]
Thus, the slope of the given line is 3.

### Step 2: Find the slope of the line perpendicular to the given line

Lines that are perpendicular have slopes that are negative reciprocals of each other. The negative reciprocal of 3 is:
\[
m = -\frac{1}{3}
\]
So, the slope of the line perpendicular to the given line is \( -\frac{1}{3} \).

### Step 3: Use the point-slope form of the equation

We know that the line we're interested in has a slope of \( -\frac{1}{3} \) and passes through the point \( (-6, 4) \). We can use the point-slope form of a line's equation:
\[
y - y_1 = m(x - x_1)
\]
Substituting \( m = -\frac{1}{3} \), \( x_1 = -6 \), and \( y_1 = 4 \):
\[
y - 4 = -\frac{1}{3}(x + 6)
\]

### Step 4: Solve for the slope-intercept form

Now, we simplify to get the slope-intercept form (\( y = mx + b \)):
\[
y - 4 = -\frac{1}{3}(x + 6)
\]
Distribute the \( -\frac{1}{3} \):
\[
y - 4 = -\frac{1}{3}x - 2
\]
Add 4 to both sides:
\[
y = -\frac{1}{3}x + 2
\]

### Final Answer:
The slope-intercept equation of the function \( f \) is:
\[
y = -\frac{1}{3}x + 2
\]

Would you like more details or have any questions?

Here are five related questions to deepen your understanding:
1. How do you find the slope of a line from two points?
2. What is the negative reciprocal of a slope, and why is it important for perpendicular lines?
3. How would the equation change if the line were parallel instead of perpendicular?
4. What is the general form of the point-slope equation of a line?
5. How can you find the intercepts of a line from its equation?

**Tip:** Perpendicular lines always have slopes that multiply to \( -1 \).

Write the slope-intercept equation of the function f whose graph satisfies the given conditions. The graph of f passes through (-6, 4) and is perpendicular to the line that has an x-intercept of 1 and a y-intercept of -3.

Discover how to find the slope-intercept equation of a line passing through a specific point and perpendicular to a given line with intercepts. This problem involves key algebraic concepts such as slopes, point-slope form, and the perpendicular slopes theorem. Ideal for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation